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a) tan x = 1

b) tany = 4 + tanx /2tanx - 3

Step-by-step explanation:

a) Given the equation

2(sinx + 2cosx) = sinx +5cosx

Open the bracket

2sinx + 4cosx =  sinx +5cosx

Collect like terms

2sinx - sinx = 5cosx - 4cosx

sinx = cos x

Divide both sides by cos x

sinx/cosx = cosx/cosx

tanx = 1

Hence the value of tan x is 1

b) Given the expression

sinxcosy + 3cosxsiny = 2sinxsiny - 4cosxcosy

We are to express tany in terms of tan x

Divide through by cosx cosy

sinxcosy/cosx cosy + 3cosxsiny/cosx cosy = 2sinxsiny/cosx cosy - 4cosxcosy/cosx cosy

sinx/cosx + 3siny/cosy = 2tanxtany - 4

tanx +3tany =  2tanxtany - 4

Collect like terms

tanx - 2tanxtany = -4 - 3tany

- 2tanxtany+3tany = -4 - tanx

tany (-2tanx+3) = -4 - tanx

tan y = -4 - tanx/-2tanx+3

tan y = -(4+tanx)/-(2tanx-3)

tany = 4 + tanx /2tanx - 3

Hence the function of tany in term of tan x is tany = 4 + tanx /2tanx - 3

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step-by-step explanation: